The Lorentz Transformation — Question 10

Question 10

Phase invariance determines the transformation of frequency and wavenumber

A plane wave has phase $\phi=kx-\omega t=k\prime x\prime-\omega\prime t\prime$. Use the inverse boost $x=\gamma(x\prime+vt\prime)$ and $t=\gamma(t\prime+vx\prime/c^2)$.

1. Collect coefficients of $x\prime$ and $t\prime$ to derive $k\prime$ and $\omega\prime$.

2. For a vacuum wave traveling along +x and $v=0.60c$, find both ratios and check its phase speed.

Question 10 - Solution

Phase invariance determines the transformation of frequency and wavenumber

(a) Method and calculation

$$\phi=\gamma(k-\omega v/c^2)x\prime-\gamma(\omega-kv)t\prime.$$$$\boxed{k\prime=\gamma(k-\omega v/c^2)},\quad\boxed{\omega\prime=\gamma(\omega-kv)}.$$

(b) Interpretation and further calculation

$$\omega=ck:\quad\boxed{k\prime/k=\omega\prime/\omega=\gamma(1-\beta)=1/2}.$$$$\boxed{\omega\prime/k\prime=c}.$$

Physical check

Frequency and wavelength change together while their product remains the vacuum propagation speed.